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Simplifying 6n2 + -6n + -420 = 0 Reorder the terms: -420 + -6n + 6n2 = 0 Solving -420 + -6n + 6n2 = 0 Solving for variable 'n'. Factor out the Greatest Common Factor (GCF), '6'. 6(-70 + -1n + n2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-70 + -1n + n2)' equal to zero and attempt to solve: Simplifying -70 + -1n + n2 = 0 Solving -70 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '70' to each side of the equation. -70 + -1n + 70 + n2 = 0 + 70 Reorder the terms: -70 + 70 + -1n + n2 = 0 + 70 Combine like terms: -70 + 70 = 0 0 + -1n + n2 = 0 + 70 -1n + n2 = 0 + 70 Combine like terms: 0 + 70 = 70 -1n + n2 = 70 The n term is -1n. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1n + 0.25 + n2 = 70 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 70 + 0.25 Combine like terms: 70 + 0.25 = 70.25 0.25 + -1n + n2 = 70.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 70.25 Calculate the square root of the right side: 8.381527307 Break this problem into two subproblems by setting (n + -0.5) equal to 8.381527307 and -8.381527307.Subproblem 1
n + -0.5 = 8.381527307 Simplifying n + -0.5 = 8.381527307 Reorder the terms: -0.5 + n = 8.381527307 Solving -0.5 + n = 8.381527307 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = 8.381527307 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 8.381527307 + 0.5 n = 8.381527307 + 0.5 Combine like terms: 8.381527307 + 0.5 = 8.881527307 n = 8.881527307 Simplifying n = 8.881527307Subproblem 2
n + -0.5 = -8.381527307 Simplifying n + -0.5 = -8.381527307 Reorder the terms: -0.5 + n = -8.381527307 Solving -0.5 + n = -8.381527307 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + n = -8.381527307 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -8.381527307 + 0.5 n = -8.381527307 + 0.5 Combine like terms: -8.381527307 + 0.5 = -7.881527307 n = -7.881527307 Simplifying n = -7.881527307Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.881527307, -7.881527307}Solution
n = {8.881527307, -7.881527307}
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